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Mirrors > Home > MPE Home > Th. List > imass1 | Structured version Visualization version GIF version |
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
Ref | Expression |
---|---|
imass1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres 6023 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | |
2 | rnss 5952 | . . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) |
4 | df-ima 5701 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
5 | df-ima 5701 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
6 | 3, 4, 5 | 3sstr4g 4040 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3962 ran crn 5689 ↾ cres 5690 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: predrelss 6359 vdwnnlem1 17028 dprdres 20062 imasnopn 23713 imasncld 23714 imasncls 23715 utoptop 24258 restutop 24261 ustuqtop3 24267 utopreg 24276 metustbl 24594 imadifxp 32620 gsumfs2d 33040 esum2d 34073 eulerpartlemmf 34356 bj-imdirco 37172 brtrclfv2 43716 frege97d 43741 frege109d 43746 frege131d 43753 hess 43769 resimass 45183 setrecsss 48931 |
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